The category of comonoids in a monoidal category. #

We define comonoids in a monoidal category C, and show that they are equivalently monoid objects in the opposite category.

We construct the monoidal structure on Comon_ C, when C is braided.

An oplax monoidal functor takes comonoid objects to comonoid objects. That is, a oplax monoidal functor F : C ⥤ D induces a functor Comon_ C ⥤ Comon_ D.

TODO #

Comonoid objects in C are "just" oplax monoidal functors from the trivial monoidal category to C.

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structure Comon_ (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] :

Type (max u₁ v₁)

A comonoid object internal to a monoidal category.

When the monoidal category is preadditive, this is also sometimes called a "coalgebra object".

X : C
The underlying object of a comonoid object.
counit : self.X ⟶ 𝟙_ C
The counit of a comonoid object.
comul : self.X ⟶ CategoryTheory.MonoidalCategory.tensorObj self.X self.X
The comultiplication morphism of a comonoid object.
counit_comul : CategoryTheory.CategoryStruct.comp self.comul (CategoryTheory.MonoidalCategory.whiskerRight self.counit self.X) = (CategoryTheory.MonoidalCategory.leftUnitor self.X).inv
comul_counit : CategoryTheory.CategoryStruct.comp self.comul (CategoryTheory.MonoidalCategory.whiskerLeft self.X self.counit) = (CategoryTheory.MonoidalCategory.rightUnitor self.X).inv
comul_assoc : CategoryTheory.CategoryStruct.comp self.comul (CategoryTheory.MonoidalCategory.whiskerLeft self.X self.comul) = CategoryTheory.CategoryStruct.comp self.comul (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerRight self.comul self.X) (CategoryTheory.MonoidalCategory.associator self.X self.X self.X).hom)

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theorem Comon_.counit_comul {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (self : Comon_ C) :

CategoryTheory.CategoryStruct.comp self.comul (CategoryTheory.MonoidalCategory.whiskerRight self.counit self.X) = (CategoryTheory.MonoidalCategory.leftUnitor self.X).inv

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theorem Comon_.comul_counit {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (self : Comon_ C) :

CategoryTheory.CategoryStruct.comp self.comul (CategoryTheory.MonoidalCategory.whiskerLeft self.X self.counit) = (CategoryTheory.MonoidalCategory.rightUnitor self.X).inv

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theorem Comon_.comul_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (self : Comon_ C) :

CategoryTheory.CategoryStruct.comp self.comul (CategoryTheory.MonoidalCategory.whiskerLeft self.X self.comul) = CategoryTheory.CategoryStruct.comp self.comul (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerRight self.comul self.X) (CategoryTheory.MonoidalCategory.associator self.X self.X self.X).hom)

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theorem Comon_.counit_comul_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (self : Comon_ C) {Z : C} (h : CategoryTheory.MonoidalCategory.tensorObj (𝟙_ C) self.X ⟶ Z) :

CategoryTheory.CategoryStruct.comp self.comul (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerRight self.counit self.X) h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.leftUnitor self.X).inv h

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theorem Comon_.comul_counit_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (self : Comon_ C) {Z : C} (h : CategoryTheory.MonoidalCategory.tensorObj self.X (𝟙_ C) ⟶ Z) :

CategoryTheory.CategoryStruct.comp self.comul (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerLeft self.X self.counit) h) = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.rightUnitor self.X).inv h

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theorem Comon_.comul_assoc_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (self : Comon_ C) {Z : C} (h : CategoryTheory.MonoidalCategory.tensorObj self.X (CategoryTheory.MonoidalCategory.tensorObj self.X self.X) ⟶ Z) :

CategoryTheory.CategoryStruct.comp self.comul (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerLeft self.X self.comul) h) = CategoryTheory.CategoryStruct.comp self.comul (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerRight self.comul self.X) (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.associator self.X self.X self.X).hom h))

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theorem Comon_.trivial_comul (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] :

(Comon_.trivial C).comul = (CategoryTheory.MonoidalCategory.leftUnitor (𝟙_ C)).inv

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theorem Comon_.trivial_X (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] :

(Comon_.trivial C).X = 𝟙_ C

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theorem Comon_.trivial_counit (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] :

(Comon_.trivial C).counit = CategoryTheory.CategoryStruct.id (𝟙_ C)

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def Comon_.trivial (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] :

Comon_ C

The trivial comonoid object. We later show this is terminal in Comon_ C.

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One or more equations did not get rendered due to their size.

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instance Comon_.instInhabited (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] :

Inhabited (Comon_ C)

Equations

Comon_.instInhabited C = { default := Comon_.trivial C }

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theorem Comon_.counit_comul_hom_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {M : Comon_ C} {Z : C} (f : M.X ⟶ Z✝) {Z : C} (h : CategoryTheory.MonoidalCategory.tensorObj (𝟙_ C) Z✝ ⟶ Z) :

CategoryTheory.CategoryStruct.comp M.comul (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.tensorHom M.counit f) h) = CategoryTheory.CategoryStruct.comp f (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.leftUnitor Z✝).inv h)

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theorem Comon_.counit_comul_hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {M : Comon_ C} {Z : C} (f : M.X ⟶ Z) :

CategoryTheory.CategoryStruct.comp M.comul (CategoryTheory.MonoidalCategory.tensorHom M.counit f) = CategoryTheory.CategoryStruct.comp f (CategoryTheory.MonoidalCategory.leftUnitor Z).inv

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theorem Comon_.comul_counit_hom_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {M : Comon_ C} {Z : C} (f : M.X ⟶ Z✝) {Z : C} (h : CategoryTheory.MonoidalCategory.tensorObj Z✝ (𝟙_ C) ⟶ Z) :

CategoryTheory.CategoryStruct.comp M.comul (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.tensorHom f M.counit) h) = CategoryTheory.CategoryStruct.comp f (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.rightUnitor Z✝).inv h)

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theorem Comon_.comul_counit_hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {M : Comon_ C} {Z : C} (f : M.X ⟶ Z) :

CategoryTheory.CategoryStruct.comp M.comul (CategoryTheory.MonoidalCategory.tensorHom f M.counit) = CategoryTheory.CategoryStruct.comp f (CategoryTheory.MonoidalCategory.rightUnitor Z).inv

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theorem Comon_.comul_assoc_flip_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {M : Comon_ C} {Z : C} (h : CategoryTheory.MonoidalCategory.tensorObj (CategoryTheory.MonoidalCategory.tensorObj M.X M.X) M.X ⟶ Z) :

CategoryTheory.CategoryStruct.comp M.comul (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerRight M.comul M.X) h) = CategoryTheory.CategoryStruct.comp M.comul (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerLeft M.X M.comul) (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.associator M.X M.X M.X).inv h))

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theorem Comon_.comul_assoc_flip {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {M : Comon_ C} :

CategoryTheory.CategoryStruct.comp M.comul (CategoryTheory.MonoidalCategory.whiskerRight M.comul M.X) = CategoryTheory.CategoryStruct.comp M.comul (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerLeft M.X M.comul) (CategoryTheory.MonoidalCategory.associator M.X M.X M.X).inv)

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theorem Comon_.Hom.ext_iff {C : Type u₁} :

∀ {inst : CategoryTheory.Category.{v₁, u₁} C} {inst_1 : CategoryTheory.MonoidalCategory C} {M N : Comon_ C} (x y : M.Hom N), x = y ↔ x.hom = y.hom

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theorem Comon_.Hom.ext {C : Type u₁} :

∀ {inst : CategoryTheory.Category.{v₁, u₁} C} {inst_1 : CategoryTheory.MonoidalCategory C} {M N : Comon_ C} (x y : M.Hom N), x.hom = y.hom → x = y

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structure Comon_.Hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (M : Comon_ C) (N : Comon_ C) :

Type v₁

A morphism of comonoid objects.

hom : M.X ⟶ N.X
The underlying morphism of a morphism of comonoid objects.
hom_counit : CategoryTheory.CategoryStruct.comp self.hom N.counit = M.counit
hom_comul : CategoryTheory.CategoryStruct.comp self.hom N.comul = CategoryTheory.CategoryStruct.comp M.comul (CategoryTheory.MonoidalCategory.tensorHom self.hom self.hom)

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theorem Comon_.Hom.hom_counit {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {M : Comon_ C} {N : Comon_ C} (self : M.Hom N) :

CategoryTheory.CategoryStruct.comp self.hom N.counit = M.counit

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theorem Comon_.Hom.hom_comul {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {M : Comon_ C} {N : Comon_ C} (self : M.Hom N) :

CategoryTheory.CategoryStruct.comp self.hom N.comul = CategoryTheory.CategoryStruct.comp M.comul (CategoryTheory.MonoidalCategory.tensorHom self.hom self.hom)

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theorem Comon_.Hom.hom_comul_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {M : Comon_ C} {N : Comon_ C} (self : M.Hom N) {Z : C} (h : CategoryTheory.MonoidalCategory.tensorObj N.X N.X ⟶ Z) :

CategoryTheory.CategoryStruct.comp self.hom (CategoryTheory.CategoryStruct.comp N.comul h) = CategoryTheory.CategoryStruct.comp M.comul (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.tensorHom self.hom self.hom) h)

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theorem Comon_.Hom.hom_counit_assoc {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {M : Comon_ C} {N : Comon_ C} (self : M.Hom N) {Z : C} (h : 𝟙_ C ⟶ Z) :

CategoryTheory.CategoryStruct.comp self.hom (CategoryTheory.CategoryStruct.comp N.counit h) = CategoryTheory.CategoryStruct.comp M.counit h

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theorem Comon_.id_hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (M : Comon_ C) :

M.id.hom = CategoryTheory.CategoryStruct.id M.X

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def Comon_.id {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (M : Comon_ C) :

M.Hom M

The identity morphism on a comonoid object.

Equations

M.id = { hom := CategoryTheory.CategoryStruct.id M.X, hom_counit := ⋯, hom_comul := ⋯ }

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instance Comon_.homInhabited {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (M : Comon_ C) :

Inhabited (M.Hom M)

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M.homInhabited = { default := M.id }

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theorem Comon_.comp_hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {M : Comon_ C} {N : Comon_ C} {O : Comon_ C} (f : M.Hom N) (g : N.Hom O) :

(Comon_.comp f g).hom = CategoryTheory.CategoryStruct.comp f.hom g.hom

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def Comon_.comp {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {M : Comon_ C} {N : Comon_ C} {O : Comon_ C} (f : M.Hom N) (g : N.Hom O) :

M.Hom O

Composition of morphisms of monoid objects.

Equations

Comon_.comp f g = { hom := CategoryTheory.CategoryStruct.comp f.hom g.hom, hom_counit := ⋯, hom_comul := ⋯ }

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instance Comon_.instCategory {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] :

CategoryTheory.Category.{v₁, max u₁ v₁} (Comon_ C)

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Comon_.instCategory = CategoryTheory.Category.mk ⋯ ⋯ ⋯

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theorem Comon_.ext {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {X : Comon_ C} {Y : Comon_ C} {f : X ⟶ Y} {g : X ⟶ Y} (w : f.hom = g.hom) :

f = g

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theorem Comon_.id_hom' {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (M : Comon_ C) :

(CategoryTheory.CategoryStruct.id M).hom = CategoryTheory.CategoryStruct.id M.X

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theorem Comon_.comp_hom' {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {M : Comon_ C} {N : Comon_ C} {K : Comon_ C} (f : M ⟶ N) (g : N ⟶ K) :

(CategoryTheory.CategoryStruct.comp f g).hom = CategoryTheory.CategoryStruct.comp f.hom g.hom

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theorem Comon_.forget_map (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] :

∀ {X Y : Comon_ C} (f : X ⟶ Y), (Comon_.forget C).map f = f.hom

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theorem Comon_.forget_obj (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (A : Comon_ C) :

(Comon_.forget C).obj A = A.X

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def Comon_.forget (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] :

CategoryTheory.Functor (Comon_ C) C

The forgetful functor from comonoid objects to the ambient category.

Equations

Comon_.forget C = { obj := fun (A : Comon_ C) => A.X, map := fun {X Y : Comon_ C} (f : X ⟶ Y) => f.hom, map_id := ⋯, map_comp := ⋯ }

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instance Comon_.forget_faithful {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] :

(Comon_.forget C).Faithful

Equations

⋯ = ⋯

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instance Comon_.instIsIsoHomOfMapForget {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {A : Comon_ C} {B : Comon_ C} (f : A ⟶ B) [e : CategoryTheory.IsIso ((Comon_.forget C).map f)] :

CategoryTheory.IsIso f.hom

Equations

⋯ = e

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instance Comon_.instReflectsIsomorphismsForget {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] :

(Comon_.forget C).ReflectsIsomorphisms

The forgetful functor from comonoid objects to the ambient category reflects isomorphisms.

Equations

⋯ = ⋯

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theorem Comon_.mkIso_hom_hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {M : Comon_ C} {N : Comon_ C} (f : M.X ≅ N.X) (f_counit : autoParam (CategoryTheory.CategoryStruct.comp f.hom N.counit = M.counit) _auto✝) (f_comul : autoParam (CategoryTheory.CategoryStruct.comp f.hom N.comul = CategoryTheory.CategoryStruct.comp M.comul (CategoryTheory.MonoidalCategory.tensorHom f.hom f.hom)) _auto✝) :

(Comon_.mkIso f f_counit f_comul).hom.hom = f.hom

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theorem Comon_.mkIso_inv_hom {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {M : Comon_ C} {N : Comon_ C} (f : M.X ≅ N.X) (f_counit : autoParam (CategoryTheory.CategoryStruct.comp f.hom N.counit = M.counit) _auto✝) (f_comul : autoParam (CategoryTheory.CategoryStruct.comp f.hom N.comul = CategoryTheory.CategoryStruct.comp M.comul (CategoryTheory.MonoidalCategory.tensorHom f.hom f.hom)) _auto✝) :

(Comon_.mkIso f f_counit f_comul).inv.hom = f.inv

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def Comon_.mkIso {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {M : Comon_ C} {N : Comon_ C} (f : M.X ≅ N.X) (f_counit : autoParam (CategoryTheory.CategoryStruct.comp f.hom N.counit = M.counit) _auto✝) (f_comul : autoParam (CategoryTheory.CategoryStruct.comp f.hom N.comul = CategoryTheory.CategoryStruct.comp M.comul (CategoryTheory.MonoidalCategory.tensorHom f.hom f.hom)) _auto✝) :

M ≅ N

Construct an isomorphism of comonoids by giving an isomorphism between the underlying objects and checking compatibility with counit and comultiplication only in the forward direction.

Equations

One or more equations did not get rendered due to their size.

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instance Comon_.uniqueHomToTrivial {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (A : Comon_ C) :

Unique (A ⟶ Comon_.trivial C)

Equations

A.uniqueHomToTrivial = { default := { hom := A.counit, hom_counit := ⋯, hom_comul := ⋯ }, uniq := ⋯ }

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instance Comon_.instHasTerminal {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] :

CategoryTheory.Limits.HasTerminal (Comon_ C)

Equations

⋯ = ⋯

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theorem Comon_.Comon_ToMon_OpOp_obj'_one (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (A : Comon_ C) :

(Comon_.Comon_ToMon_OpOp_obj' C A).one = A.counit.op

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@[simp]

theorem Comon_.Comon_ToMon_OpOp_obj'_mul (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (A : Comon_ C) :

(Comon_.Comon_ToMon_OpOp_obj' C A).mul = A.comul.op

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theorem Comon_.Comon_ToMon_OpOp_obj'_X (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (A : Comon_ C) :

(Comon_.Comon_ToMon_OpOp_obj' C A).X = Opposite.op A.X

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def Comon_.Comon_ToMon_OpOp_obj' (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (A : Comon_ C) :

Mon_ Cᵒᵖ

Turn a comonoid object into a monoid object in the opposite category.

Equations

Comon_.Comon_ToMon_OpOp_obj' C A = { X := Opposite.op A.X, one := A.counit.op, mul := A.comul.op, one_mul := ⋯, mul_one := ⋯, mul_assoc := ⋯ }

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theorem Comon_.Comon_ToMon_OpOp_obj (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (A : Comon_ C) :

(Comon_.Comon_ToMon_OpOp C).obj A = Opposite.op (Comon_.Comon_ToMon_OpOp_obj' C A)

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theorem Comon_.Comon_ToMon_OpOp_map (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] :

∀ {X Y : Comon_ C} (f : X ⟶ Y), (Comon_.Comon_ToMon_OpOp C).map f = Opposite.op { hom := f.hom.op, one_hom := ⋯, mul_hom := ⋯ }

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def Comon_.Comon_ToMon_OpOp (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] :

CategoryTheory.Functor (Comon_ C) (Mon_ Cᵒᵖ)ᵒᵖ

The contravariant functor turning comonoid objects into monoid objects in the opposite category.

Equations

One or more equations did not get rendered due to their size.

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theorem Comon_.Mon_OpOpToComon_obj'_X (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (A : Mon_ Cᵒᵖ) :

(Comon_.Mon_OpOpToComon_obj' C A).X = Opposite.unop A.X

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@[simp]

theorem Comon_.Mon_OpOpToComon_obj'_comul (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (A : Mon_ Cᵒᵖ) :

(Comon_.Mon_OpOpToComon_obj' C A).comul = A.mul.unop

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@[simp]

theorem Comon_.Mon_OpOpToComon_obj'_counit (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (A : Mon_ Cᵒᵖ) :

(Comon_.Mon_OpOpToComon_obj' C A).counit = A.one.unop

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def Comon_.Mon_OpOpToComon_obj' (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (A : Mon_ Cᵒᵖ) :

Comon_ C

Turn a monoid object in the opposite category into a comonoid object.

Equations

Comon_.Mon_OpOpToComon_obj' C A = { X := Opposite.unop A.X, counit := A.one.unop, comul := A.mul.unop, counit_comul := ⋯, comul_counit := ⋯, comul_assoc := ⋯ }

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theorem Comon_.Mon_OpOpToComon__obj (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (A : (Mon_ Cᵒᵖ)ᵒᵖ) :

(Comon_.Mon_OpOpToComon_ C).obj A = Comon_.Mon_OpOpToComon_obj' C (Opposite.unop A)

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theorem Comon_.Mon_OpOpToComon__map_hom (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] :

∀ {X Y : (Mon_ Cᵒᵖ)ᵒᵖ} (f : X ⟶ Y), ((Comon_.Mon_OpOpToComon_ C).map f).hom = f.unop.hom.unop

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def Comon_.Mon_OpOpToComon_ (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] :

CategoryTheory.Functor (Mon_ Cᵒᵖ)ᵒᵖ (Comon_ C)

The contravariant functor turning monoid objects in the opposite category into comonoid objects.

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theorem Comon_.Comon_EquivMon_OpOp_unitIso (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] :

(Comon_.Comon_EquivMon_OpOp C).unitIso = CategoryTheory.NatIso.ofComponents (fun (x : Comon_ C) => CategoryTheory.Iso.refl ((CategoryTheory.Functor.id (Comon_ C)).obj x)) ⋯

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theorem Comon_.Comon_EquivMon_OpOp_counitIso (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] :

(Comon_.Comon_EquivMon_OpOp C).counitIso = CategoryTheory.NatIso.ofComponents (fun (x : (Mon_ Cᵒᵖ)ᵒᵖ) => CategoryTheory.Iso.refl (((Comon_.Mon_OpOpToComon_ C).comp (Comon_.Comon_ToMon_OpOp C)).obj x)) ⋯

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theorem Comon_.Comon_EquivMon_OpOp_functor (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] :

(Comon_.Comon_EquivMon_OpOp C).functor = Comon_.Comon_ToMon_OpOp C

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theorem Comon_.Comon_EquivMon_OpOp_inverse (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] :

(Comon_.Comon_EquivMon_OpOp C).inverse = Comon_.Mon_OpOpToComon_ C

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def Comon_.Comon_EquivMon_OpOp (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] :

Comon_ C ≌ (Mon_ Cᵒᵖ)ᵒᵖ

Comonoid objects are contravariantly equivalent to monoid objects in the opposite category.

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instance Comon_.instMonoidalCategoryOfBraidedCategory (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] :

CategoryTheory.MonoidalCategory (Comon_ C)

Comonoid objects in a braided category form a monoidal category.

This definition is via transporting back and forth to monoids in the opposite category,

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Comon_.instMonoidalCategoryOfBraidedCategory C = CategoryTheory.Monoidal.transport (Comon_.Comon_EquivMon_OpOp C).symm

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theorem Comon_.tensorObj_X (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (A : Comon_ C) (B : Comon_ C) :

(CategoryTheory.MonoidalCategory.tensorObj A B).X = CategoryTheory.MonoidalCategory.tensorObj A.X B.X

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theorem Comon_.tensorObj_counit (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (A : Comon_ C) (B : Comon_ C) :

(CategoryTheory.MonoidalCategory.tensorObj A B).counit = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.tensorHom A.counit B.counit) (CategoryTheory.MonoidalCategory.leftUnitor (𝟙_ C)).hom

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theorem Comon_.tensorObj_comul' (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (A : Comon_ C) (B : Comon_ C) :

(CategoryTheory.MonoidalCategory.tensorObj A B).comul = CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.tensorHom A.comul B.comul) (CategoryTheory.tensor_μ Cᵒᵖ (Opposite.op A.X, Opposite.op B.X) (Opposite.op A.X, Opposite.op B.X)).unop

Preliminary statement of the comultiplication for a tensor product of comonoids. This version is the definitional equality provided by transport, and not quite as good as the version provided in tensorObj_comul below.

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theorem Comon_.tensorObj_comul (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.BraidedCategory C] (A : Comon_ C) (B : Comon_ C) :

The comultiplication on the tensor product of two comonoids is the tensor product of the comultiplications followed by the tensor strength (to shuffle the factors back into order).

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